Nearest Neighbor Density Functional Estimation from Inverse Laplace Transform

Abstract

A new approach to L2-consistent estimation of a general density functional using k-nearest neighbor distances is proposed, where the functional under consideration is in the form of the expectation of some function f of the densities at each point. The estimator is designed to be asymptotically unbiased, using the convergence of the normalized volume of a k-nearest neighbor ball to a Gamma distribution in the large-sample limit, and naturally involves the inverse Laplace transform of a scaled version of the function f. Some instantiations of the proposed estimator recover existing k-nearest neighbor based estimators of Shannon and Rényi entropies and Kullback–Leibler and Rényi divergences, and discover new consistent estimators for many other functionals such as logarithmic entropies and divergences. The L2-consistency of the proposed estimator is established for a broad class of densities for general functionals, and the convergence rate in mean squared error is established as a function of the sample size for smooth, bounded densities.